FormalConjectures.ErdosProblems.«108»

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import FormalConjecturesUtil

Erdős Problem 108

Reference: erdosproblems.com/108


universe u


namespace Erdos108


open Erdos108

For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_108 :
    answer(sorry) ↔ ∀ r ≥ 4, ∀ k ≥ (2 : ℕ), ∃ (f : ℕ),
    ∀ (V : Type u) (G : SimpleGraph V) (_ : Nonempty V)
      (hchro : f ≤ SimpleGraph.chromaticNumber G),
    ∃ (H : G.Subgraph), (SimpleGraph.girth H.coe ≥ r) ∧
    (SimpleGraph.chromaticNumber H.coe ≥ k) := by⊢ True ↔
  ∀ r ≥ 4,
    ∀ k ≥ 2,
      ∃ f,
        ∀ (V : Type u) (G : SimpleGraph V),
          Nonempty V → ↑f ≤ G.chromaticNumber → ∃ H, H.coe.girth ≥ r ∧ H.coe.chromaticNumber ≥ ↑k
  sorryAll goals completed! 🐙


-- TODO: Proof for the case r=4 and statement for the infinite case

end Erdos108